15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 MHD Equilibrium and Stability of Tokamaks and FP Systems with 3D Helical Cores W. A. Cooper Ecole Polytechnique Fédérale de Lausanne, Association EUATOM-Confédération Suisse, Centre de echerches en Physique des Plasmas, CH115 Lausanne, Switzerland. Collaborators EPFL/CPP: J. P. Graves, O. Sauter, A. Pochelon, L. Villard Consorzio FX: D. Terranova, M. Gobbin, P. Martin, L. Marrelli, I. Predebon ONL: S. P. Hirshman Culham Science Centre: I. T. Chapman W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 1
MOTIVATION Manifestation of internal 3D structures in axisymmetric devices: SHAx states in FX-mod. Lorenzini et al., Nature Physics 5 (29) 57 Snakes in JET A. Weller et al., Phys. ev. Lett. 59 (1987) 233 Disappearance of sawteeth but continuous dominantly n = 1 mode in TCV at high elongation and current H. eimerdes et al., Plasma Phys. Control. Fusion 48 (26) 1621 Y. Camenen et al., Nucl. Fusion 47 (27) 586 Change of sawteeth from kink-like to quasi-interchange-like with plasma shaping in DIII-D E. A. Lazarus et al., Plasma Phys. Contr. Fusion 48 (26) L65 Long-lived saturated modes in MAST I. T. Chapman et al., Nucl. Fusion 5 (21) 45997 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 2
THEOY EVIEW Analytic studies of ideal nonlinearly saturated m = 1, n = 1 modes Avinash,.J. Hastie, J.B. Taylor, Phys. ev. Lett. 59 (1987) 2647 M.N. Bussac,. Pellat, Phys. ev. Lett. 59 (1987) 265 F.L. Waelbroeck, Phys. Fluids B 59 (1989) 499 Pellet deposition in magnetic island on q = 1 surface J.A. Wesson, Plasma Phys. Control. Fusion B 37 (1995) A337 Large scale simulations of nonlinearly saturated MHD instability L.A. Charlton et al., Phys. Fluids B 59 (1989) 798 H. Lütjens, J.F. Luciani, J. Comput. Phys. 227 (28) 6944 Bifurcated equilibria due to ballooning modes with the NSTAB code P. Garabedian, Proc. Natl. Acad. Sci. USA 13 (26) 19232 FX-mod SHAx MHD equilibria D. Terranova et al., in Plasma Phys. Contr. Fusion 52 (21) Bifurcated tokamak equilibria similar to a saturated internal kink W.A. Cooper et al., Phys. ev. Lett. 15 (21) 353 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 3
OUTLINE We investigate the proposition that the instability structures observed in the experiments constitute in reality new equilibrium states with 3D character; an axisymmetric boundary is imposed 3D magnetohydrodynamic (MHD) fixed boundary equilibria with imposed nested flux surfaces are investigated with: VMEC2 S. P. Hirshman, O. Betancourt, J. Comput. Phys. 96 (1991) 99 ANIMEC W. A. Cooper et al., Comput. Phys. Commun. 18 (29) 1524 MHD equilibria with 3D internal helical structures computed for TCV, ITE (hybrid scenario), MAST, JET (Snakes) and FX-mod Linear ideal MHD stability computed with TEPSICHOE for FXmod D. V. Anderson et al., Int. J. Supercomp. Appl. 4 (199) 33 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 4
Magnetohydrodynamic Equilibria 3D Impose nested magnetic surfaces and single magnetic axis Minimise energy of the system W = ( B d 3 2 x + p (s, B) ) 2µ Γ 1 Solve inverse equilibrium problem : = (s, u, v), = (s, u, v). Variation of the energy dw dt = s=1 [ dsdudv F [ dudv t + F (p + B2 2µ )( u t + F λ ] λ t t u )] t W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 5
Magnetohydrodynamic Equilibria 3D The MHD forces are F = F z = u [σ gb u (B )] + v [σ gb v (B )] [ )] (p + B2 + [ )] (p + B2 u s 2µ s u 2µ g ) + [(p + B2 σ 2 (B v ) 2] 2µ u [σ gb u (B )] + v [σ gb v (B )] + [ )] (p + B2 [ )] (p + B2 u s 2µ s u 2µ The λ force equation minimises the spectral width and corresponds to the binormal projection of the momentum balance at the equilibrium state. [ F λ = Φ (σbv ) (s) (σb ] u) u v For isotropic pressure p = p = p and σ = 1/µ. W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 6
Magnetohydrodynamic Equilibria 3D Use Fourier decomposition in the periodic angular variables u and v and a special finite difference scheme for the radial discretisation An accelerated steepest descent method is applied with matrix preconditioning to obtain the equilibrium state The radial force balance is a diagnostic of the accuracy of the equilibrium state in this approach Fs 1 p Φ = (s) Φ σbv ι(s) σbu (s) s B s g s g denotes a flux surface average A = L (2π) 2 2π/L dv 2π du ga(s, u, v) This model is implemented in the ANIMEC code, an anisotropic pressure extension of the VMEC2 code. L is the number of toroidal field periods. W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 7
Boundaries for TCV, ITE, MAST and JET Fixed axisymmetric boundary equilibrium studies are explored. TCV boundary description: b =.8 +.2 cos u +.6 cos 2u; b =.48 sin u MAST, JET boundary: b = + a cos(u + δ sin u + τ sin 2u); b = Ea sin u MAST: =.9m, a =.54m, E = 1.744, δ =.3985, τ =.198 JET: = 2.96m, a = 1.25m, E = 1.68, δ =.3, τ =.5.4.3.2.1 TCV 3 2 1 ITE 1.8.6.4.2 MAST 2 1.5 1.5 JET.1.2.3.4.5.6.8 1 1 2 3 4 5 6 7 8.2.4.6.8 1.5 1 1.5.5 1 1.5 2 1.5 2 2.5 3 3.5 4 4.5 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 8
Bifurcated equilibria in TCV Selection of q-profiles that yield bifurcated equilibria in TCV. Boundary description: b =.8 +.2 cos u +.6 cos 2u, b =.48 sin u. q =(.5+s 1.1s 4 ) 1, (.7+.7s s 4 ) 1, (.9+.2s.8s 6 ) 1, (.98.7s 9 ) 1. 4 3.5 q = 1/ι 3 2.5 2 1.5 1.5.2.4.6.8 1 s W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 9
CPP TCV toroidal flux contours at various cross sections TCV toroidal magnetic flux contours with prescribed ι =.9+.2s.8s6 profile. Axisymmetric v=π v= v = π/3 v = 2π/3 Helical W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 1
Helical equilibrium structures in TCV Helical axis displacement and energy difference as function of q min. q(s) = [.9 + α 1 s (.6 + α 1 )s 6 ] 1, α 1 =.16.32. 1 (); 1 () µ (W hel W axi ) x 1 5 1 () ; 1 ().2.15.1.5.5.1.94.96.98 1 1.2 1.4 q min µ (W hel W axi ).5 1 1.5 2.96.98 1 1.2 q min W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 11
TCV helical structures at finite β Flat core pressure and q-profile with weak shear reversal q(s) = (.9 +.3s s 4 ) 1. p(s) = p (1 +.41227s 2 14.1988s 4 + 29.6253s 6 22.9512s 8 + 6.1152s 1 ). p, q-profiles 1 (); 1 () p(s)/p() ; q(s) 2 1.5 1.5.2.4.6.8 1 s 1 () ; 1 ().2.15.1.5.5.1.1.2.3 <β> W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 12
ITE equilibrium profiles Prescribe mass profile and toroidal current profile. p(s) M(s)[Φ (s)] Γ Equilibria have toroidal current 13 14MA, B t = 4.6T, β 2.9% q-profile q-profile range for helical states 4.5 4 3.5 1.1 1.8 1.6 1.4 13.1MA q 3 2.5 2 1.5 q 1.2 1.98.96.94.92 13.8MA 1.2.4.6.8 1 s.9.45.5.55.6.65.7 s W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 13
CPP ITE pressure contours at various cross sections Contours of constant pressure of an ITE hybrid scenario equilibrium with 13.3M A toroidal plasma current v= W. Anthony Cooper, CPP/EPFL v = π/3 v = 2π/3 v=π 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 14
Convergence of solutions Define helical excursion parameter δ H = 2 1 (s = ) + 1 2 (s = ) a Convergence of δ H with N r = number of radial grid points.45.4.35.3.25 δ H.2.15.1.5.5.2.4.6.82 1 1.2 1.4 1.6 1/N r x 1 4 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 15
Helical core ITE equilibria Variation of the helical axis distortion parameter δ H with respect to the toroidal current and corresponding variation with respect to q min. δ H versus current δ H versus q min.4.4.35.35.3.3.25.25 δ H.2 δ H.2.15.15.1.1.5.5 13 13.2 13.4 13.6 13.8 14 TOOIDAL CUENT.97.98.99 1 1.1 1.2 q min W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 16
CPP ITE pressure contours as a function of plasma current Contours of constant pressure at the cross section with toroidal angle v = 2π/3 13.1M A 13.2M A 13.3M A 13.4M A 13.5M A 13.6M A 13.7M A 13.8M A W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 17
Finite β ITE scan The ITE hybrid scenario is projected in the range 12 14MA with B t = 5.3T. With a sharper core concentrated toroidal current at 12M A, the variation of the helical axis distortion parameter δ H with respect to β is computed..25.2 δ H.15.1.1.2.3.4 <β> W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 18
MAST pressure contours at various cross sections Contours of constant pressure of a MAST equilibrium 1.8 v = (a) 1.8 v = π/3 (b) 1.8 v = 2π/3 (c) 1.8 v = π (d).6.6.6.6.4.4.4.4.2.2.2.2.2.2.2.2.4.4.4.4.6.6.6.6.8 1 v=.5 1 1.5.8 1 v=π/3.5 1 1.5.8 1 v=2π/3.5 1 1.5.8 1 v=π.5 1 1.5 This serves as a model for long-lived modes in MAST -I.T. Chapman et al., Nucl. Fusion 5 (21) 45997 Saturated internal kinks also observed on NSTX -J.E. Menard et al., Nucl. Fusion 45 (25) 539 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 19
JET snake equilibrium profiles Prescribe mass profile and toroidal current profile. Equilibrium has toroidal current 3.85MA, B t = 3.1T, β 2.3% pressure profile q-profile pressure.35.3.25.2.15.1.5.2.4.6.8 1 s q 9 8 7 6 5 4 3 2 1.2.4.6.8 1 s W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 2
JET pressure, mod-b contours at various cross sections CPP Contours of constant pressure and mod-b of a JET snake equilibrium v= v=π v = π/3 v = 2π/3 2 (b) 2 (c) 2 (d) 1.5 1.5 1 1 1 1.5.5.5.5 1.5 1.5 2 (a).5.5.5.5 1 1 1 1 1.5 1.5 1.5 1.5 2 2 v= 2 3 4 W. Anthony Cooper, CPP/EPFL 2 3 v=π/3 4 2 2 3 v=2π/3 4 2 v=π 2 3 4 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 21
JET snake equilibrium structure Standard view of a snake (with variation with respect to toroidal angle in lieu of time in the simulation). experiment (courtesy of A. Weller) ANIMEC simulation W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 22
FX-mod toroidal flux and j B/B 2 contours Contours of constant toroidal flux (VMEC coordinates) and j B/B 2 (Boozer coordinates) in a FX-mod SHAx equilibrium state.4 v =.4 v = π/21.4 v = 2π/21.4 v = π/7.3.3.3.3.2.2.2.2.1.1.1.1.1.1.1.1.2.2.2.2.3.3.3.3.4 φ=.4 v=π/21.4 v=2π/21.4 v=π/7 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4.4 φ =.4 φ = π/21.4 φ = 2π/21.4 φ = π/7.3.3.3.3.2.2.2.2.1.1.1.1.1.1.1.1.2.2.2.2.3.3.3.3.4 φ=.4 φ=π/21.4 φ=2π/21.4 φ=π/7 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 1.6 1.8 2 2.2 2.4 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 23
FX-mod toroidal flux and mod-b contours CPP Contours of constant toroidal flux and mod-b in the Boozer coordinate frame for the FX-mod SHAx equilibrium state φ=.4 φ = π/21.4 φ = 2π/21.3.2.2.2.2.1.1.1.1.3.3.1.1.1.1.2.2.2.2.3.3.4 1.6 1.8 φ= 2.2 2 2.4 φ= W. Anthony Cooper, CPP/EPFL.4 1.6.3 1.8 φ=π/21 2.2 2.4 2.4 1.6.3 1.8 φ=2π/21 2.2 2.4 2 φ = π/21 φ = π/7.4.3.4 φ = 2π/21.4 1.6 1.8 φ=π/7 2.2 2.4 2 φ = π/7 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 24
Mod-B 2 Fourier amplitudes in FX-mod Dominant Fourier amplitudes of B 2 in the FX-mod SHAx equilibrium state 1.6 1.4 m/n=/ 1.2 1.8 B 2 mn.6.4.2 m/n=1/14.2 m/n=/7 m/n=1/7 m/n=1/.4.2.4.6.8 1 s W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 25
Linear Ideal MHD Stability Internal potential energy δw p = 1 2 d 3 x [C 2 + Γp ξ 2 D ξ s 2] Use Boozer coordinates ξ = gξ s (B s) θ φ + η B 2 + Fourier decomposition in angular variables [ J(s) Φ (s)b 2η µ ] B ξ s (s, θ, φ) = l s q lx l (s)sin(m l θ n l φ + ) η(s, θ, φ) = l Y l (s)cos(m l θ n l φ + ) A hybrid finite element method is applied for the radial discretisation. In the 3D TEPSICHOE stability code, COOL finite elements based on variable order Legendre polynomials have been implemented. The order of the polynomial is labelled with p. W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 26
MHD stability in FX-mod Ideal MHD stability with respect to n = 8 family of modes (w/a = 1.99).14.12.1 λ 4.13 1 6 λ 4.6 1 4 λ 2.55 1 1 q=1/8 q.8.6.4.2.2.4.6.8 1 s W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 27
Fourier Amplitudes of ξ s The three leading Fourier amplitudes of the radial component of the displacement vector ξ s mn with w/a = 1.198 and p = 5 for core q /q < for core q /q >.2.15 core dq/ds < m/n=1/8.12.1.8 core dq/ds > m/n=1/8 ξ s mn.1.5 m/n=2/15 m/n=1/6.5.2.4.6.8 1 S ξ s mn.6.4.2 m/n=1/6.2 m/n=2/15.2.4 S.6.8 1 W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 28
Wall Stabilisation Effects λ for core q /q < for core q /q > 1 1 1 1 p=1 1 2 p=5 1 2 1 3 λ λ 1 3 1 4 1 5 1 4 1 1.5 2 2.5 3 w/a 1 6 1 1.5 2 2.5 3 w/a W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 29
The Instability Driving Terms The magnetohydrodynamic instability driving terms correspond to ballooning/interchange (δw BI ) and kink (δw J ) structures δw BI = 1 2 + δw J = 1 2 1 1 ds 1 1 ds ds ds 2π L s 2π L s 2π L s 2π L s dφ dφ dφ dφ 2π 2π 2π 2π [ gp dθp (s) + ψ (s)j(s) Φ (s)i(s) (s) ] g (ξ s ) 2 s dθp (s) B s B 2ξs ( gb ξ s ) [ gb 2( ) j B dθ s 2 B 2 ( ) j B (ξ s ) 2 B 2 ( j B dθ B 2 B 2 ) h s ξ s ( gb ξ s ) ] + ψ (s)φ (s) Φ (s)ψ (s) W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 3
Perturbed Internal Energy Profiles The kink mode driving energy δw J and the ballooning/interchange driving energy δw BI profiles with core q /q > at different conducting wall positions δw J δw BI δ W J.2.4.6.8 1 1.2 1.4 1.6 x 1 3 w/a=1.198 ( 1 1 ) w/a=2.976 w/a=1.395 1.8.2.4.6.8 1 S δ W BI 1 2 3 4 5 6 7 8 1 x 1 6 w/a=1.198 w/a=2.976 w/a=1.395 9.2.4.6.8 1 S W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 31
Structure of δw J The structure of the kink driving energy δw J for core q /q < and w/a = 2.976 at three cross sections covering half of a field period (1/14th of the torus) W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 32
Summary and Conclusions equilibrium Nominally axisymmetric eversed Field Pinch and Tokamak systems can develop MHD equilibrium bifurcations that lead to the formation of core helical structures. In FX-mod, the development of a SHAx equilibrium state is computed with a seven-fold toroidally periodic structure when q in the core 1/7. In Tokamak devices, reversed magnetic shear (sometimes just very flat extended low shear) with q min 1 can trigger a bifurcated solution with a core helical structure similar to a saturated m = 1, n = 1 internal kink. We have computed these 3D core helical states in TCV, ITE, MAST and JET. The JET snake phenomenon is reproduced with our model. W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 33
Summary and Conclusions MHD stability Brief periods in which the relatively quiescent SHAx state in FXmod relaxes to a turbulent multiple helicity state are observed. Lorenzini et al., Nature Physics 5 (29) 57. We have investigated the ideal MHD stability of SHAx equilibria by examining mode structures that break the m = 1, n = 7 periodicity of the system. Either a drop in q well below 1/8 or the disappearance of core shear reversal triggers ideal MHD modes dominated by m = 1, n = 8 coupled with m = 2, n = 15 components. For w/a > 1.2, global unstable kink mode structures are triggered. These kink modes are driven by the Ohmic current. The drive for ballooning and Pfirsch-Schlüter current modes is very weak. The dominant mode structure is a nonresonant m = 1, n = 6 component and is basically internal with a finite edge amplitude. Coupling with toroidal sidebands constitutes a significant contribution. W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 34
Summary and Conclusions In tokamaks, the issue of MHD stability is complicated by the fact that any mode to be investigated is in principle also a component of the equilibrium state. Possible exceptions: local ballooning/mercier modes, external kink modes, stellarator-symmetry breaking modes. The 3D helical core equilibrium states in nominally axisymmetric systems that have been obtained constitute a paradigm shift for which the tools developed for stellarators in MHD stability, kinetic stability, drift orbits, wave propagation or heating, neoclassical transport, gyrokinetics, etc become applicable and necessary to more accurately evaluate magnetic confinement physics phenomena. The constraint of nested magnetic flux surfaces and absence of X- points in our model preclude the generation of equilibrium states with magnetic islands. Saturated tearing modes could be investigated with SIESTA, PIES, HINT or SPEC. W. Anthony Cooper, CPP/EPFL 15th Workshop on MHD Stability Control, Madison, WI, USA, November 15-17, 21 35